Final Project - The Astroid
by
Lydia Silva and Brianne Yokoyama
What
is an Astroid?
A hypocycloid is the curve traced by a point on a circle of radius r which is being rolled along the inside circumference of another circle of a larger radius R
A cusp
is defined to be a “point at which two branches of a curve meet such that the
tangents of each branch are equal.”
An astroid is a specialized case of the general hypocycloid family of curves. When you have two circles, one with a radius of R, and the other with a radius of r, where r is ¼ the length of R, the resulting curve is a star-like shape with four points or cusps. This curve was given the simple name astroid in 1836. The word astroid is the term from which the modern word asteroid is derived.
The relation between
the number of cusps and the ratio of lengths R to r is simple. The formation of
a cusp occurs every time the smaller circle completes a full rotation along the
inside of the larger circle. To roll all the way around the inside of the larger
circle, it must make R/r rotations. In the case of the astroid, the smaller
circle which we will call circle 1 has a circumference that is precisely ¼ that
of the larger circle, circle 2. Therefore, as circle 1 is rolled along the
inside of the circle 2, it must make four revolutions before returning to its
original position, resulting in a curve with four cusps.
The
equation for the astroid in Cartesian coordinates is x2/3+y2/3=a2/3
For
a simple parametrization of the curve you can substitute u for x1/3 and
v for y1/3 and r for a1/3 and
the equation becomes u2 +v2=r2. Using the
parameters u=rcos(t) and v=rsin(t) and substituting our x,y, and z values, we
get the following parametric equations:
x1/3=
a1/3 cos(t)
(x1/3)3=(
a1/3 cos(t))3
x=acos3(t)
y1/3=
a1/3 sin(t)
(y1/3)3=(
a1/3 sin(t))3
y=asin3(t)
The Danish astronomer
Ole Roemer, famed for his discovery that light has a finite velocity, first
discovered the astroid curve in 1674 in studying applications of the properties
of hypocycloids in relation to gear teeth. Scientists and mathematicians Johann
Bernoulli, Gottfried Leibniz, and Jean Le Rond d’Alembert performed further
investigations of the astroid. In 1725, David Bernoulli discovered the double
generation property of the astroid. His work showed that the astroid curved
traced within a circle of radius R is created by two circles, not just one. The
first we already know has a radius of ¼R, but the second will have a radius of
¾R.
You can use astroids to make many interesting patterns such as the Astroid Diamond shown below.
References:
Lee, Xha. “Astroid”, Visual Dictionary of Specialized Curves, 2004, 13 May, 2007 http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html
“Ole Roemer” Love to Know Website, 29 Aug. 2007. 13 May 2007 <http://www.1911encyclopedia.org/Ole_Roemer>
Wassenar, Jan. “Astroid”, Mathematical Curves, Roullette. 13 March 2005. 13 May 2007http://www.2dcurves.com/roulette/roulettea.html
Weisstein, Eric W. "Astroid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Astroid.html
O’Hanen, Benjamin and Matthew Wisan. “The Asteroids: Special Planes Curve.” 15 May 2006. 13 May 2007 <http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/sp06/mattben/RealWebsite/AsteroidPaper.html>